On a class of Hamiltonian polytopes
type: Item
instance of: scholarly article
title: On a class of Hamiltonian polytopes
zbMATH Open document ID: 0655.05045
DOI: 10.1016/0012-365X(88)90103-3
author: Peter Mihók
author: Stanislav Jendroľ
published in: Discrete Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Let \({\mathcal S}(p,q)\) denote the class of simple 3-polytopal graphs all of whose edges are incident with two p-gons or a p-gon and a q-gon, \(p\neq q\), p,q\(\geq 3\). In this paper it is shown that all graphs of the class \({\mathcal S}(5,q)\) for \(3\leq q\leq 12\), \(q\neq 5\) are Hamiltonian, using the following proposition due to \textit{S. K. Stein}: A simple 3-polytopal graph G is Hamiltonian if and only if its dual \(G^*\) has point- arboricity equal to 2 [Bull. Am. Math. Soc. 76, 805-806 (1970; Zbl 0194.560)]. The theorem in the present paper supplements the known results of \textit{P. R. Goodey} that all graphs of the classes \({\mathcal G}(3,6)\) and \({\mathcal G}(4,6)\) are Hamiltonian, where \({\mathcal G}(p,q)\) denotes the class of 3-connected 3-valent planar graphs, i.e. simple 3-polytopal graphs, all of whose faces are p-gons and q-gons, \(p<q\), \(p\geq 3\) [Isr. J. Math. 22, 52-56 (1975; Zbl 0317.05114); J. Graph Theory 1, 181-185 (1977; Zbl 0379.05037)].
Mathematics Subject Classification ID: 05C45
zbMATH DE Number: 4068918
zbMATH Keywords: Hamiltonian polytopes
zbMATH Keywords: 3-connected 3-valent planar graphs
zbMATH Keywords: point- arboricity
reviewed by: Ioan Tomescu
MaRDI profile type: Publication
cites work: The Point-Arboricity of Planar Graphs
cites work: Hamiltonian circuits in polytopes with even sided faces
cites work: A class of Hamiltonian polytopes
cites work: Q5547252
cites work: On the simplicial 3-polytopes with only two types of edges
cites work: Simple 3-polytopal graphs with edges of only two types and shortness coefficients
cites work: Regular planar graphs with faces of only two types and shortness parameters
cites work: $B$-sets and coloring problems
cites work: Non-Hamiltonian simple 3-polytopes having just two types of faces
description: scientific article; zbMATH DE number 4068918